Исследование 132 костей человека показало, что их размеры не подчиняются закону Бенфорда

132 Human Bones Were Measured to Test a Famous Number Law, and the Raw Lengths Failed It Two orthopedic surgeons in Hangzhou, China, pulled X-rays of 132 bones from three adult patients and asked an unusual question. Do the numbers describing human bones behave like the numbers describing rivers, stock prices, and international trade? The answer turned out to be complicated, and the researchers are careful to say it is not a medical finding. Raw bone lengths and perimeters clearly did not match the mathematical pattern they were tested against. Projected areas and mathematically transformed values came closer. The study was published in the International Journal of General Medicine by Yunlong Zhi and Cheng Ji of the Affiliated Hangzhou First People's Hospital, School of Medicine, Westlake University. Nothing in this paper tells readers anything about their own bones, their fracture risk, or their skeletal health. It is a curiosity, and the authors frame it that way in their own conclusions. A Curiosity Test, Not a Clinical Finding Benford's Law describes a quirk in how numbers begin. In many real-world datasets, the leading digit is not evenly distributed. The digit 1 shows up first about 30.1 percent of the time, the digit 2 about 17.6 percent, and so on down to about 4.6 percent for the digit 9. The pattern turns up in trade figures, natural hazard data, and even word frequency rankings across five languages. The catch is that the pattern depends heavily on the shape of the data. Datasets spanning several orders of magnitude tend to show it. Datasets squeezed into a narrow numerical range often do not. Human long bones are anatomically constrained by definition, mostly falling within one or two orders of magnitude, so the authors designed the work as a description rather than a verdict on whether bones obey the law. The researchers screened four patients who had undergone X-rays of at least half their long bones after trauma evaluations between January 2019 and December 2023. One was excluded for hallux valgus, a bunion deformity. That left three: a 31-year-old man, a 66-year-old man, and a 55-year-old woman, none with fractures, tumors, or deformities. Numbers Behind the Comparison The team measured the length, perimeter, and projected area of 44 long bones per patient, yielding 132 observations. They also calculated squared and cubed length values, since two-dimensional and three-dimensional quantities relate geometrically to squared and cubed linear dimensions. Raw lengths diverged detectably from the expected distribution (p = 0.001). The mismatch was driven largely by an oversupply of numbers starting with 2, which appeared 29.5 percent of the time, compared with the expected 17.6 percent. Perimeters also diverged, again at 0.001, but in the opposite direction, with only 3 percent of values starting with 2. Projected areas and squared lengths did not show statistically significant deviations from the expected distribution, with p-values of 0.260 and 0.292, respectively. Cubed lengths came closest, at 0.910. Mean absolute deviation, a plain measure of how far observed proportions sat from expected ones, followed the same order: 0.042 for length, 0.037 for perimeter, 0.024 for area, 0.026 for squared length, and 0.014 for cubed length. The authors immediately flag a problem with their own most striking result. Cubing a number expands its numerical range, and expanding that range can, in turn, push a dataset toward a Benford-like pattern. The improvement may be arithmetic rather than biological. Three Patients Cannot Speak for a Population The limitations section is unusually candid, and it deserves more attention than the headline result. Three patients is a very small sample. Although 132 bones were analyzed, they were not independent observations because they came from the same three skeletons. A single person's proportions influence dozens of data points. Standard front-to-back X-rays also cannot measure true three-dimensional surface area, volume, or mass. What the team called projected area is a flattened shadow, not a real surface. The authors state explicitly that these projections should not be interpreted as genuine three-dimensional measurements. They also caution against over-reading the p-values. A result above 0.05 was treated only as failure to reject the hypothesis, not as proof of conformity, and the chi-square test cannot quantify how well data fit. The findings, they write, should be read as "descriptive evidence of digit-distribution characteristics in long-bone morphometric data." The work was funded by the Medical and Health Research Project of Zhejiang Province, and the authors declare no conflicts of interest. It went through single anonymous peer review with two reviewer comments. Digit Patterns Already Have a Job in Medicine The reason this line of inquiry is not purely whimsical is that first-digit analysis has a practical use in research integrity. Investigators have applied Benford-style checks to look for fabricated or manipulated numbers in scientific datasets, since invented figures often fail to reproduce natural digit patterns. The approach has been explored in histology measurements and in European COVID-19 case reporting. The authors suggest that reference patterns for skeletal measurements might eventually help with data quality assessment or validation of morphometric databases. They are equally clear about what the results must not be used for. These findings, they write, should not serve as diagnostic criteria for skeletal disease or as evidence that any individual measurement is abnormal. For a general reader, the honest takeaway is modest. A mathematical pattern that governs many natural datasets does not cleanly govern the dimensions of human bones, probably because bones do not vary across enough orders of magnitude for the pattern to emerge. Whether that changes with larger, more diverse samples and true three-dimensional scanning is an open question the authors say future studies should address, with analyses stratified by sex, age, height, ethnicity, and skeletal disease. Key Questions Answered What is Benford's Law? It describes a tendency in many real-world datasets for smaller digits to appear first more often than larger ones. The digit 1 leads roughly 30 percent of the time and the digit 9 roughly 5 percent. It appears in trade data, natural hazard measurements, and some biomedical datasets. What did this study actually find? Raw long-bone lengths and perimeters deviated detectably from the expected Benford distribution. Projected areas and squared or cubed length values matched more closely. Cubed lengths matched best, though the authors note that cubing expands numerical range and may create that effect artificially. Does this say anything about bone health? No. The authors state directly that the results should not be used as diagnostic criteria for skeletal disease or as evidence that an individual measurement is abnormal. Nothing in the paper applies to personal medical decisions. How large was the sample? Three adult patients, contributing 132 long bone measurements. The authors acknowledge that the bones were not independent observations because they came from the same three people, which limits the extent to which the findings can be generalized. Why would researchers study this at all? First-digit analysis is used as a screening tool to assess data quality and potential fabrication in scientific datasets. Establishing what normal digit patterns look like in skeletal measurements could support validation of morphometric databases. What would make the findings stronger? Larger and more diverse samples, direct three-dimensional CT measurement of bone volume rather than flat X-ray projections, and analyses broken out by sex, age, height, ethnicity, and skeletal pathology. Published by Medicaldaily.com